Problem 56 - College Algebra Essentials

Question

find and simplify the difference quotient $$ \frac{f(x+h)-f(x)}{h}, h \neq 0 $$ for the given function. $$ f(x)=7 x $$

Step-by-Step Solution

Answer

The simplified difference quotient for the function \( f(x) = 7x \) is \( 7 \).

Step 1: Substitute for f(x + h)

First, find \( f(x + h) \) by substituting \( x + h \) into the given function \( f(x) = 7x \). So, \( f(x + h) = 7(x + h) = 7x + 7h \).

Step 2: Substitute into Difference Quotient

Next, substitute \( f(x + h) \) and \( f(x) \) into the difference quotient formula. The difference quotient formula becomes \( \frac{{7x + 7h - 7x}}{h} \).

Step 3: Simplify the Numerator

The difference quotient then becomes \( \frac{7h}{h} \). Simplify by cancelling out the \( h \) in the numerator and the denominator, noting that \( h \neq 0 \).

Step 4: Final Simplification

The final step of simplification gives \( 7 \), which is the constant function derivative.

Key Concepts

Function Substitution

When dealing with difference quotients, function substitution is the first critical step. Imagine you are given a function, such as \( f(x) = 7x \). The process requires you to replace `x` with `(x + h)` in the function. This substitution aims to find \( f(x + h) \), which represents the function's value at a slightly shifted input.
To do this, substitute `(x + h)` into the function \( f(x) \). In our example:

• Substitute `(x + h)` for `x`: \( f(x+h) = 7(x + h) \).
• Distribute the `7`: \( f(x + h) = 7x + 7h \).

This step lays the foundation for understanding how changes in `x` affect the function's value and is essential for finding the difference quotient.

Numerator Simplification

After substituting, the next step is simplifying the numerator of the difference quotient. In essence, we have to subtract \( f(x) \) from \( f(x + h) \). Let's see how this works with our function.
Using the previous step's result:

• \( f(x + h) = 7x + 7h \)
• Subtract \( f(x) = 7x \) to get the difference: \( (7x + 7h) - 7x \)
• Result after subtraction: \( 7h \)

The key takeaway here is to carefully eliminate terms that appear both positively and negatively. In this case, the \( 7x \) terms cancel out, simplifying the expression. This simplification is vital because it reduces complexity, leading us closer to the derivative.

Derivative Calculation

The final step in examining the difference quotient involves calculating the derivative, which essentially measures the rate at which the function changes. After obtaining \( 7h \) from the previous simplification, we need to complete the calculation.
Divide by `h`:

• The difference quotient becomes \( \frac{7h}{h} \).
• Cancelation occurs: \( h \) is present in both numerator and denominator.
• This gives \( 7 \), but remember: \( h eq 0 \).

The outcome of \( 7 \) is crucial because it represents the derivative of \( f(x) = 7x \). A constant result means the function changes at a consistent rate, precisely `7`, for any value of `x`. Understanding this step is key to grasping how derivatives reveal the behavior of functions.